Problem: Rewrite the function by completing the square. $g(x)= x^{2} +15 x +54$ $g(x)=$
Solution: $\begin{aligned} g(x)&= x^2 +15 x +54 \\\\ &= \left(x^2 +15 x\right) +54 \end{aligned}$ Now we want to complete $x^2 +15 x$ into a perfect square. To do that, we should add $\left(\dfrac{{15}}{2}\right)^2={\dfrac{225}{4}}$ to it: $x^2{+15}x+{\dfrac{225}{4}}=\left(x +\dfrac{15}{2}\right)^2$ We add ${\dfrac{225}{4}}$ inside the parentheses, and subtract ${1}\cdot{\dfrac{225}{4}}$ outside them, to keep the expression equivalent. $\begin{aligned} &\phantom{=} \left(x^2 +15 x\right) +54 \\\\ &=\left(x^2 +15 x+{\dfrac{225}{4}}\right) +54 -{1}\cdot{\dfrac{225}{4}} \\\\ &= \left(x +\dfrac{15}{2}\right)^2 +54 -\dfrac{225}{4} \\\\ &= \left(x +\dfrac{15}{2}\right)^2 -\dfrac{9}{4} \end{aligned}$ In conclusion, the function after completing the square is written as: $g(x)= \left(x +\dfrac{15}{2}\right)^2 -\dfrac{9}{4}$